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In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. Essentially a jet group describes how a Taylor polynomial transforms under changes of coordinate systems (or, equivalently, diffeomorphisms). The ''k''-th order jet group ''G''''n''''k'' consists of jets of smooth diffeomorphisms φ: R''n'' → R''n'' such that φ(0)=0.〔.〕 The following is a more precise definition of the jet group. Let ''k'' ≥ 2. The gradient of a function ''f:'' R''k'' → R can be interpreted as a section of the cotangent bundle of R''K'' given by ''df:'' R''k'' → ''T *''R''k''. Similarly, derivatives of order up to ''m'' are sections of the jet bundle ''Jm''(R''k'') = R''k'' × ''W'', where : Here R * is the dual vector space to R, and ''Si'' denotes the ''i''-th symmetric power. A function ''f:'' R''k'' → R has a prolongation ''jmf'': R''k'' → ''Jm''(R''k'') defined at each point ''p'' ∈ R''k'' by placing the ''i''-th partials of ''f'' at ''p'' in the ''Si''((R *)''k'') component of ''W''. Consider a point . There is a unique polynomial ''fp'' in ''k'' variables and of order ''m'' such that ''p'' is in the image of ''jmfp''. That is, . The differential data ''x′'' may be transferred to lie over another point ''y'' ∈ R''n'' as ''jmfp(y)'' , the partials of ''fp'' over ''y''. Provide ''Jm''(R''n'') with a group structure by taking : With this group structure, ''Jm''(R''n'') is a Carnot group of class ''m'' + 1. Because of the properties of jets under function composition, ''G''''n''''k'' is a Lie group. The jet group is a semidirect product of the general linear group and a connected, simply connected nilpotent Lie group. It is also in fact an algebraic group, since the composition involves only polynomial operations. ==Notes== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Jet group」の詳細全文を読む スポンサード リンク
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